Apologies - I just intended to describe and
differentiate two very different types of "sound absorption" in my post - both deal with vibrational energy, but two very different applications:
1) materials designed to block sound energy (eg. acoustic foam)
2) materials designed to dampen vibrations carried through a surface (eg. damping material)
As far as what materials work best for either category - I'm sure there are studies in each category, anywhere from formal white papers to casual comparisons of "which product is best" (which themselves often vary in quality). But to get you started - this initial differentiation alone could get you un-stuck, if that's where you were stuck in your research.
It is frequently confusing - these are two very
different categories of application, that all too frequently get commingled in discussions. That was my point, essentially.
I know this little bit, that frequencies are a factor, for both:
For sound transmission - bass is harder to block than treble. That's true no matter the material. I'm sure simple density plays a role there, but at least at higher frequencies there are materials that can absorb sound energy, converting it into heat (which is actually what the previously mentioned box stuffing does).
For vibration damping - the whole goal IS to convert higher frequency vibrations into lower frequency, adding mass is usually the technique, but there's also materials that try to actively absorb - and especially for those, I'm sure frequency becomes a factor.
As far as measuring reverb or echo... that is a great question.
I'm thinking you could actually use a 2 channel oscilloscope - with a frequency generator going to one channel (I suppose that would be optional, if you only have a 1 channel, like I do - but there are inexpensive PC-based (USB) oscilloscopes out there that do two channel) and also to your amplifiers, and a microphone connected to the other. In that way, you could send impulse bursts into your amplifier and speaker/speakers in your listening space... so trace 1 would show you the reference signal - perfectly clean with no echo, including where it ended - and your trace 2 would show you the measured response at the position of the microphone, including echo. You would have to factor in sound transmission time (speed of sound) to properly align the two signals, but that sounds like a pretty scientific approach to me - but that's just my intuition. It would be time consuming, because I think you'd want to burst individual frequencies, working your way across the frequency spectrum, to find resonances and echoes. I'm not aware of a reference rig or specific tool, other than the visual comparison that would offer.
Even in the highest levels of car audio, most people take an approach like this:
"Hey, something is vibrating in my car when I'm listening really loud."
"Let's take a look - if it's something that's loose or vibrating or rubbing, we can secure it down. We can also use sound damping material to stop panels from vibrating. Can you put something on that makes it do what you want to stop?"